QUANTUM WEYL ALGEBRAS and DEFORMATIONS of U(G)

Paciﬁc Journal of Mathematics

QUANTUM WEYL ALGEBRAS AND DEFORMATIONS OF U(g)

NAIHUAN JINGAND JAMES ZHANG

Volume 171 No. 2 December 1995 PACIFIC JOURNAL OF MATHEMATICS Vol. 171, No. 2, 1995

QUANTUM WEYL ALGEBRAS AND DEFORMATIONS OF U(G)

NAIHUAN JING AND JAMES ZHANG

We construct new deformations of the universal enveloping algebras from the quantum Weyl algebras for any R-matrix. Our new algebra (in the case of g = SI2) is a noncommutative and noncocommutative bialgebra (i.e. quantum semi- group) with its localization being a Hopf algebra (i.e. quantum group). The ring structure and representation theory of our algebra are studied in the case of SI2.

Introduction.

Quantum groups are usually considered to be examples of ^-deformations of universal enveloping algebras of simple Lie algebras or their restricted dual algebras. The simplest example is the Drinfeld-Jimbo quantum group λ C/g(sl2), which is an associative algebra over C(q) generated by e,/, k,k~ subject to the following relations:

ke = q2ek, kf = q~2fk, k-k-1 ef-fe =q-q-1 One sees that the element k does not stay in the same level as the elements e and / do, since k is actually an exponential of the original element h in the Cartan subalgebra. There do exist deformations to deform all generators into the same level, for example, Sklyanin algebras. However it is still uncertain if Sklyanin algebra is a bialgebra or not. Using the quantum Weyl algebras discussed in [WZ, GZ], we will construct new deformations for the enveloping algebras which have such a nice property. Our algebra in the case of sl2 is an associative algebra generated by e, /, h with the relations:

qhe — eh — 2e, hf - qfh = -2/,

ef-qfe^h + ^h2.

437 438 NAIHUAN JING & JAMES ZHANG

Moreover the algebra is a bialgebra with the comultiplication

A(x) = x ® 1 + 1 ® x + -(1 - q)h ® a; with x being any of the basic generators e, /, Λ. Its localization is a Hopf algebra. Clearly our algebra is noncommutative and noncocommutative or a quantum group according to Drinfeld. Our new deformations will enjoy all the nice properties carried by the enveloping algebras such as having the same global, Krull, Gelfand-Kirillov dimensions as well as the same Hubert series for the associated graded rings. Our deformation is similar to Witten's algebra ([W], see also [L]) in one aspect that the Casimir element in our algebra skew-commutes with other generators. We also remark that the homogenization of both Witten's algebra and our algebra belong to a single family of algebras studied in a recent work [LSV] from a different perspective. The idea behind our construction is very simple and natural. We start from the derivatives and multiplication operators on a skew polynomial ring associated to a Hecke R-matrix. They form a quantum Weyl algebra as discussed by [GZ]. Then we go on to consider the subalgebra generated by the quadratic elements to obtain a new deformation.

We study the ring structure for our new algebras in the case of sl2. The representation theory is also worked out for both generic and singular cases of q. It is not surprising that the representation theory of the new algebra is quite similar to that of the Drinfeld-Jimbo quantum algebra Uq(g). Our approach is general in the sense that our method works for a wide class of R-matrices. In other words, we just reverified the profound principle that a meaningful R-matrix can give a deformation to the simple Lie algebras [FRT]. Recently Ding and Prenkel [DF] have introduced another kind of quantum Weyl and Clifford algebras from the L-operator approach of the quantum inverse scattering method and use them to give representations of the quantum algebras. Their work differs from ours in forming a different quadratic expression for the basic generators to give realization of the usual Drinfeld- Jimbo quantum algebras, while our result is to focus on obtaining a new quantum algebra structure.

We will use the usual symbol Uq(g) to denote our new deformed algebra unless indicated otherwise. NEW DEFORMATIONS 439

1. Quantum Weyl Algebras.

Let R = (rfj) be a Hecke R-matrix. Namely, R is a n2 by n2 matrix satisfying the Hecke relation and braid relation:

1 2 ι (1.1) (R-q)(R + q- )=0 or R = 9Λ - q~ R + 1;

The braid relation (1.2) means that for all i,j,g and all u,k,f

kf ul st (Λ *\\ V r r r — Y^ r*t uk If \L'°) Z_^rlt 'is' jg — Z^'ij'sl ' tg l,s,t l,s,t summed over the repeated indices Z,s,£ and here i? = (rfj).

Given such an R-matrix, we can define the quantum Weyl algebra An(R) as follows. Quantum Weyl algebras were first considered in [WZ]. Here we follow [GZ] to define the quantum Weyl algebra An(R). The generators are 1 n ι xl5 , xn and 9 , , 9 , and each d corresponds to the partial derivative in the ordinary Weyl algebra. The relations are

(1.4.1)

(1.4.2) &Xj ••

(1.4.3) S,t for all i and j. We also assume that R has a skew inverse matrix in the following sense: there is P = (pf*) such that ΣpfMϊ = <5ifc<^u = Σriίpί ϊj which is always true in the considered cases. If such P did not exist then ι ι we could not write xkd in terms of d Xj as we see from (1.4). We refer the reader to [GZ] for a detailed explanation why the relations (1.4) are natural analog of the usual defining relations for the classical Weyl algebra realized on the polynomial algebra in n variables. In the classical case, there is a well-known homomorphism from the en- j veloping algebra C/(gln) to the n-th Weyl algebra An by sending ei3 to Xid . (This homomorphism is not injective.) Motivated by this classical fact we are going to construct a quantum enveloping algebra generated by the quantum elements XidK Let's denote the element x$ by e^, then we have the following commutation relations for e\.

Proposition 1.1. For all n, z, /, k 440 NAIHUAN JING & JAMES ZHANG

(b)

= Σ rfc + Q-1 Σ (Σ tfβί ) (Σ rteή - <τx Σ 4 (Σ »

Proof. We only give a proof for (a) (and (b) can be handled similarly). It follows from (1.4.2) that

ι X& xkd = Xi

Then we have (Σ r"ίe0 • 4 = Σ '.I^+1Σ r^^ = Σ * 2W + Σr:;^ fa*' = Σ ^"^i^ + Σ rTjήtr^x^d'd1 by (1.4.1) = Σ r?k*i& + Σ rϋrj^x^d'd1 by (1.2)

>^ + Σ ήlr^Xvq-1 (Vx, - Jj) ^ by (1.4.2)

= Σ r%*i& + I"' Σ r&rZxv#z.& - = Σ rSeJ + 4-1 Σ (Σ C<) (Σ rίW.

Remark. We conjecture that the relations in (a) are equivalent to the relations in (b). For simplicity we introduce some notations:

u 1 u u u Of course, by (1.4.2), b z = q' (d xz - δ z). Then the matrix (b z)nxn is denoted by B. cul = V^ ruieι- = V^r^x θ'

and the matrix (c^.)nxn is denoted by C\ It is easy to see that ΣC% = B.

zk / j zi k / j NEW DEFORMATIONS 441

and the matrix (d^)nxn is denoted by D\. It is easy to see that ΣD* = B.

By using these notations we can re-write the relations in (a) and (b) as follows: u ι 2 ul ι u ι ι 1 u (a) h ep — π~ Lr —n- fi pe 4- π~ V r^c hυ W °z k ~ H zk H °z k ^ Q λ^ zk j for which we use R2 = (q — q~ι)R + 1. In the matrix form,

where / is the identity matrix. (b) 4K = q-2d% ~ q-'δ^A + q-ι ΣC^, or, e{B = q-'BDf + q-2D{ - q^Ie?. By (a)

By(b)

= q~xBB + q~2B - q~xI • T.

As a consequence B • T = T • B and (B + q~γT){B - qT) = 0.

Corollary 1.2. The element T = Σe\ commutes with elements xkd{ for all k,l in the quantum Weyl algebra An{R). Proof. This follows from the equality TB = BT and the fact that e\ =

2. Definition of C/,(gln).

We are going to construct a deformation of the enveloping algebra t/(gln) out of the subalgebra generated by the quadratic elements Xid}. We can not use all the relations in Proposition 1.1 (a), since in the classical case, the map from £7(gln) or ί7(sln) for n > 2 to the n-th Weyl algebra An by 442 NAIHUAN JING & JAMES ZHANG sending e^ to Xidj is not injective. We need to find out what are the proper relations for Uq(gln).

Let R be the R-matrix for the standard quantum group GLn(q). As a tensor, we may write R as

As a matrix R is given by R = (r*j) and rjj = 9, Γ ) = 1 for all i, j, and r*j = # — g"1 for all i < j. The quantum Weyl algebra constructed from this particular R-matrix is denoted by An(q).

It follows from (1.4) in Section 1 that the relations of An(q) are the following:

(2.1.1) XiXj = qXjXi,Vi < j;

(2.1.2) d'dj =ιj

(2.1.3) drx^

2 l 2 j (2.1.4) d'xi = 1 + q Xid + (q - 1) Σx3d ,ML

If we denote \ij\ = 1 for i < j, \ij\ = 0 for i = j, and |ΐj| = —1 for i > j, then (2.1.1) and (2.1.2) can be re-written as

(2.1.1') xixj = qMχjxiyiJ,

(2.1.27) d'dj = q-liJldjd\Vi,j.

1 z j Let us find the relations between e\dk = x^^ x^d and e^e] = Xjfc9 Xid .

They should be the relations for Uq(gln). Proposition 2.2. The quadratic elements e\ have the following relations:

(<Γ2 -

2 2 =e\- et + {q - 1) £ e\e\ - (q - NEW DEFORMATIONS 443

Proof. We only show the last one. Prom the relations (2.1) it follows that

T •H-' T ΓJ n Ύ H T • rr'

1 2 j 2 2 2 s j = Xid - q Xjd + (q - 1) V ^^9*9* - g (g - 1) V xjXsd d

2 j 2 t i = x^ - q Xjd + (q - 1) Σ xtd xid

2 j 2 - q Xjd + (q - tf t>3

.Pxi& + Q2 Σ s>j

2 j j 2 +q • q^Xjtf • XjΘ - Xjθ - (q -

1 2 j 2 2 = Xid - q Xjd + (q - lfad* + (q - 1)

2 s j j 1 - (q - 1) i Σ xsd xjd + Xjd • Xjd + {}>s>i = Xid' - Xjd1 + (q2 - 1) Σ *t&Xi& - (<12 - 1) Σ χsdsXjdj.

We will take this as our definition of new deformation of C/(gln) and denote it also by C/g(gln). The elements e{ are the analog of the Weyl generators for C/(gln). However the element Γ = £] e\ is not central when n > 2 though it commutes with all the basic elements ej, |i — j\ < ±1. If we add all relations in Proposition 1.1, then T will be central, but it is not natural to do so. This phenomenon is not strange: we know that there is no unique way to define all root elements ( elements like e*) in the Drinfeld-Jimbo quantum algebras, while in our situation we have the analog of the Weyl generators but we do not have the commutativity of T in Uq(gln).

3. Quantum algebras Uq(g}2) and Uq(sl2)> We will study our algebra in detail for the case n = 2 in this section. As a direct consequence of the relations (2.2) we have 444 NAIHUAN JING & JAMES ZHANG

Proposition 3.1. The algebra Uq(gl(2)) generated by ej , i,j = 1,2 Λαs following complete relations:

11 11 _ 1 , /2 1\-lz>2 ^2 1 12 — 2 ' W / 2 2'

1 2 2 1 ~~~ 5

2 2 2 2 2 β?βj - 9 e^ = el - e + (1 - <7 )(e2 ) ,

P2p2 _ Π2 2 2 _ p2 ele2 " e2el — eU

2 2 i 2 2 — 2*

Notice that the element T = e\ + e\ belongs to the center. Replacing g2 by ς and using the standard notations: e = e2,/ = e^Λ = e\ — e2 and α = e} + e2, we obtain

Corollary 3.2. TΛe algebra C/g(gl2) «5 generated by e,f,a,h subject to the following relations:

— eh = 2e, = -2/,

ef -qfe = a

We define the algebra Ug(s\2) to be the quotient algebra of the above

Uq(g\2) modulo the central element a. We will still use the same symbols for the generators for E/g(sl2), namely, the algebra is generated by the elements e, /, h subject to relations (3.2) with a = 0. It is clear that they specialize to the usual enveloping algebras C/(gl2) or Ϊ7(sl2) when q goes to 1. Our algebra is a bialgebra with the following comultiplication:

Δ(a ) = x®l + l®x + -(1 - q)h ® x Δ where x is the generator /ι, e, and /, and Δ is extended to the whole algebra by linearity and multiplicativity. The counit is the morphism given by

e(h) = e(β) = e(f) = 0, c(l) = 1.

In fact it is easy to check that the mappings Δ and e are well-defined and satisfy all the properties of a bialgebra. We can also enlarge our algebra into NEW DEFORMATIONS 445 a Hopf algebra via its localization by the element 1 + |(1 — q)h using the antipode determined by

x for x being any of the basic generators /ι,e, /. It is easy to check that 1 -f |(1 — q)h is a normal and group-like element, and it is also a regular element because ί7g(sl2) is a domain (see Theorem 3.7). The noncommutativity and noncocommutativity of the bialgebra structure means that our algebra is a quantum (semi)group in the sense of Drinfeld.

The algebra Uq(sl2) has a Casimir element given by

(3.3) = 2qfe + h + \h2

= 2ef-h + \h2.

The Casimir element g-commutes with the generators in the following sense:

Lemma 3.4.

eC = qCe,

hC = Ch.

Proof. Let's check one of them, say the first one:

eC - qCe = e [2qfe + h+ -hΛ - q [2ef -h+ ^hΛ e

= (eft + qhe) + \{eh2 - q2h2e) = 0. Δ D

Remark. The Casimir element also plays an important role in the following associated homogenization ring Hq(s\2):

qhe — eh = 2e£, t is central, hf - qfh = -2ft,

2 ef -qfe = ht + ^—^h . 446 NAIHUAN JING & JAMES ZHANG

When 9 = 1, this ring is studied by [LS]. It is easy to check that Hq(sl2) is

an Ore extension k[k, ί][e,

and derivative δ2, and that the elements h — 2t/(q — 1) and C are normal. In [LSV] similar type of algebras are considered and they also found such properties for their algebras. To study the ring structure of our new algebra we have the following result.

Proposition 3.5. (1) The algebra Uq(gl2) is an Ore extension

k[h,a][e,σ1][f,σ2,δ2]

for some proper automorphisms σλlσ2 and σ2-deriυation δ2. The ideal generated by elements h, α, e, / is a maximal ideal of co-dimension 1;

(2) The algebra Uq(sl2) is an Ore extension k[h][e, σ[][f, σ2, δ2] for some proper

automorphisms σ[,σ'2 and σ2-derivation δ'2. The ideal generated by elements h,e,f is a maximal ideal of co-dimension 1.

Proof. It is routine to verify the statements from the relations (3.2). For

example, σx(α) = a and σι(h) = qh — 2. D

The following result is clear by Proposition 3.5.

Corollary 3.6. The algebra Uq(sl2) (resp. Uq(gl2)) has a PBW-type basis consisting of elements fihjek with i,j,k E Z+ (resp. fιhjekaι with + i,j,k,l G Z ). Consequently, the associated graded ring gr(Uq(sl2)) (resp. gr(Uq(gl2))) has the same Hilbert series as its classical counterpart. In fact,

both gr(Uq(sl2)) and gr(Uq(g)l2)) are skew polynomial rings.

As a consequence of Proposition 3.5 and Corollary 3.6, we get

Theorem 3.7. The algebra Uq(sl2) (resp. Uq(gl2)) is a noetherian domain of Gelfand-Kirillov dimension, Krull dimension, and global dimension 3 (resp. 4).

Proof. By Proposition 3.5 and [MR, Thm 1.2.9], both algebras are noetherian domains. The assertion about Gelfand-Kirillov dimension is clear because the associated graded rings are skew polynomial rings. For simplicity the dimension in this proof will mean either Krull dimension or global dimension. By Proposition 3.5 and [MR, Prop 6.5.4, Thm 7.5.3] it follows that the dimension of Uq(s\2) (resp. (79(gl2)) is at most 3 (resp. 4). Note that α,

1 + |(1 — q)h, ef — fe is a regular sequence of £/q(gl2) with the factor ring isomorphic to k[x,x~λ]. By [MR, Lemma 6.3.10, Thm 7.3.5], the dimension of Uq(sl2) (resp. £7g(gl2)) is at least 3 (resp. 4). D NEW DEFORMATIONS 447

Remark. Note that Hq(sl2)/(t-l) 9* Uq{s\2) and Hq(sl2)/(t) 9* gr(Uq(sl2)). 4 Thus Hq(sl2) has Hubert series 1/(1 — £) , and it is a Noetherian domain, a maximal order, Auslander-regular ring of dimension 4 with the Cohen- Macaulay property (see [LS]).

4. Representation theory of Uq(sl2).

Our algebra Uq(gln) is realized on the infinite dimensional quantum Weyl algebra An(R), which serves as a natural defining module. In this section we will study finite dimensional modules for our algebra Uq(sl2). Assume q is a complex number. For μ G C, we define the /i-weight space for a [/^-module V as

Vμ = {υ e V\h.υ = μυ} whose elements are called weight vectors with weight μ. It is easy to check that fVμ C Vqμ-2,eVμ C Vq-iμ+2q-i. As in the classical case we define the Borel subalgebra U(b) as the subalgebra generated by the element e and h. A λ-weight vector is called a highest weight vector if it is annihilated by the element e and dimVχ = 1.

A [/g(sl2)-module V is called a highest weight module if it is generated by a highest weight vector. Note that when λ = -^j, the weight space V\ is stablized under the actions of e and /, which is not the case we want to consider. From now on we assume any considered weight is not equal to -^ except in some remarks. Let Cλ = C ® υ\ be the 1-dimensional module for the Borel subalgebra U(b) generated by e, h. Here

h.vx — \υx, e.vλ = 0.

The Verma module V(λ) = Uq ®c/(&) Cλ is a highest weight module for Uq with the highest weight λ. Lemma 4.1. Fix λ G C. The submodules ofV(X) are generated by vectors j 1±q fυχ with some j such that λ = —2 1_ . Proof. We will use the following g-integers:

As a direct computation we have that hf=qJΓh-2[j]f\

efj - qjfje = -[ ' 448 NAIHUAN JING & JAMES ZHANG

Set v = 1 ®V\. From the above computations It is clear that a submodule of V(X) has a linear basis consisting of elements of the form fjv. Suppose j is the minimum integer for a non-zero fjυ inside the submodule, then e kills the element fjv. Thus

efjv = [j] (~\j - 1] + λ^"1 + \λ2qj-l(l - ?)) ΓXv = 0,

which implies that \\2qj-ι(l - q) + \qj~λ - [j - 1] = 0. D

The above result implies that the Verma module has a unique maximal j submodule Uqf v for some j EN (or {0} if no such j exists). The quotient module is a simple irreducible [/^-module with the highest weight λ, denoted as L(λ).

Theorem 4.2. Suppose q is not a root of 1. For each half positive integer

s G N/2, there exist exactly two irreducible highest weight Uq-modules of dimension 2s + 1, namely V = L(X) where (1) The highest weights are λ = — 2ι^~ (2) Relative to h, V is the direct sum of weight spaces

and dimVμ = 1;

(3) The action of Uq is given by the following formulae:

evj = -[-

= —2 - q J where Vj = f^vχ/\j]\, and j = 0,1, , 25.

Proof. Let V be an irreducible [/^-module of dimension 25 + 1. Pick an eigen- vector or weight vector υ for h. The irreducibility implies that V is spanned by the vectors of the form {e*/j.^} It follows from the finite dimensionality that there exists a highest weight vector v\ for V. Thus V = Uqvχ. Set v Jυ j j = f \/[JV J = 0,1, . Since υά has weight q λ - 2[j]J = 0,1, , all the weights qjλ — 2[j] are different due to λ ψ -^. Thus the vectors Vj are linearly independent. Therefore it must happen that υ2s+ι = 0. w Applying e to t>25+i, ^ have that NEW DEFORMATIONS 449 thus λ = - l — q For λ = 2^- it is easy to check that the weights for are actually i\ - 2[j] = q -21:7~*+i for 0 < j < 2s. Then

Finally it is easy to verify that the formulae in (3) do give a representation for Uq. D

Remark. (1) When q —> 1, the module L(X) with λ = — 2-j£— specializes to the standard (2s + l)-dimensional module for f/(sl2), while the other one does not have a specialization. (2) The 1-dimensional modules are easily seen to be determined by

/ \ L * £ ± (α) /l-^I'e/--(^riji' (6) e = / = 0, either h = 0, or h =

Proposition 4.3. For eαc/t n > 0, there is an n-dimensional right Uq(sl2)- module Wn satisfying

(1) 0 C Wi C W2 C C Wn-ι C Wn and a// factor modules WnjWn-λ (which are 1-dimensional) are isomorphic to Wι, and

(2) Wn is not isomorphic to a direct sum of two submodules, in particular

Wn is not completely reducible. Hence Uq(sl2) is not "semi-simple".

n Proof. For any n > 0, let Wn = C . The n x n matrix ring acts on Wn naturally. Denote

/I 0 0 0 ••• 0 0\ 1 1 0 0 ••• 0 0 0 1 1 0 ••• 0 0 0 0 1 1 0 0

V 0 0 0 0 1 1/

/n = -- and 9-1" where In is the identity n x n-matrix. One can easily check that en,/n,/ιn satisfy the relations of C/g(sl2). Hence there is a unique ring homomorphism 450 NAIHUAN JING & JAMES ZHANG

from Uq(sl2) to Endk(Wn) mapping e to en, / to fn and h to hn, and Wn is right Uq(sl2)-modnle. For example we have 2

Xi e = Xf en = Xi + Xi—i ) and Xi - h = xι - hn = #$ 91 where #i, ,#n is a basis of Wn. Prom this we see that en\Wm = em, e n|wm = /m and Λn|wm = hm where Wm is identified with the subspace

ΈJILi k%i of Wn for all m < n. Hence Wm is a submodule of Wn. It is easy to check the the factor module Wn/Wm is isomorphic to Wn-m. By the definition of en, the vector space Wn can not be written as a direct sum of two en-invariant subspaces. Hence Wn is not a direct sum of two submodules. D

Now let us focus on the case when the parameter q is a root of unity. In the remaining part of this section we assume that q is an n-th primitive root 2πm of 1, i.e., q = e /^ for n > 2 and (ra,n) = 1 for positive integers ra,n.

n n Lemma 4.4. The elements e and f are in the center of the algebra Uq. Proof. It is a consequence of the commutation relations and [n] = 0. (Note that hn does not belong to the center.) D

We are going to consider the representations of the quotient algebra Uq n n of the algebra Uq by the ideal generated by e , f .

Theorem 4.5. Every finite dimensional simple Uq-module is of dimension

(1) For each 2 < d < n, there are exactly two simple d-dimensional Uq- 1 g modules which are precisely L(λ) with λ = —2 ^ 1_

(2) There are infinitely many simple n-dimensional Uq-modules which are

jL(λ); where X is not a zero of the polynomials

lλ^-\i-q) + χq^-y-i], j = o, . ,n-i.

Proof. Let V be a simple module of dimension d. By a similar argument as at the beginning of the proof for Theorem (4.2) (for generic q) it follows that V is a highest weight module. Say V is of highest weight λ with the highest weight vector υ\, then V is determined by

e.VX = 0, h.Vχ = \Vχ j d e.f vx φ 0, j = 1, , d - 1, and e.f vx = 0. NEW DEFORMATIONS 451

j Then the set of vectors fυχ, j = 0, , d — 1 form a basis for V , and d < n due to [n] — 0. If dim V < n, then the last condition in the above says that

1 d ι e/V = ~[d\ (-[d - 1] + λ^ + \\Y-Hl - qή f -vχ = 0, which means that λ = — 2—%-. . Then l-q

Therefore V ^ L(X) and the action is described in Theorem 4.2. n Suppose the dimension of V is n. The condition e.f vχ = 0 is satisfied automatically due to [n] = 0. We only worry about the first set of require- ments:

which completes the proof. D

In the end we will demonstrate that our algebra has (reducible) indecomposable modules of dimension 2n when q is an nth primitive root, which is similar to the observation made by [K] for the Drinfeld-Jimbo algebra

Uq(sl2). However, we only have finite number of indecomposable modules for each case versus the situation for the Drinfeld-Jimbo algebra Uq(sl2).

Theorem 4.6. For each 1 < s < n — 1 there exist indecomposable Uq- modules V of dimension 2n such that

(1) The module V is a direct sum of its weight spaces Vμ with

-(n+β-l)/2 l -r Q-(n+s-3)/2 J -r- (n+s-l)/2 , 222, ^, 2^ μ = ~ l-q l-q l-q

ι q with the highest weight — 2 ^ χ_ (2) The dimensions of the weight spaces are 2 for the weight string

μ = -2-^X , -2-^- , , -2-^ , l-q l-q l-q and 1 otherwise; (3) We can choose υ to be a highest weight vector and w to be a weight

/n s-ί-l) / 2 q vector in Vμ with μ — —2^ λ_ independent from v such that the 452 NAIHUAN JING & JAMES ZHANG

action is given by

j j x e.fυ = -\j][-(n + s - j)]f ' v7 j = 0, , n - 1,

1 π: a3-{n+s-l)/2 j h.fv = -2 \_g f v] i = 0, - , ,n-l,

e./'u; = gV^-1^ - [j][-(n - β - j)]/'"-1*;, j = 1, , n - 1,

1 IΠ o3-(ns-l)/2 h.f'w = -2—^ /'w; i = 0. n - 1. 1-9

Proof. Let V be a module satisfying conditions (1) and (2). We want to show that the module structure is given by formulae in (3). Take a weight vector v with weight λ = — 2ι^q ^J . Then

e.fjv = ~[j][-(n + s- j)]fj~ιv φ 0 for j φ s.

1=Fςr Since dim Vμ = 2 for μ — —2 ^_ , there exists a weight vector w G Vμ not in the kernel of e such that

e.w —

s which implies that f v and w form a basis for the Vμ. Then we have

e.fw = qjfi+s~lv - [j][-{n - s - j)]fj~ιw, j = 1, , n - 1 which implies that e.fn~sw = q~sfn~1v, so vectors f-^w^j = 0, , n — 1 are independent (nonzero). It is clear that vectors fjw and fs+jv generate the ι q3 s weight space Vμ for μ = — 2^ ^~ , j = 0, , j = n — s +1. So vectors fjv,fjw,j=0, - , n — 1 form a basis for V. Finally it is easy to check that the formulae in (3) do define a representation of Uq. The indecomposability of the module is clear from the defining formulae. D

5. Connection to Drinfeld-Jimbo quantum groups.

The Drinfeld-Jimbo quantum algebra Uq(sl2) was first introduced by Kulish- Reshetikhin and Sklyanin in early 80's as an associative algebra generated NEW DEFORMATIONS 453

by e, /, h over the ring of power series in t with the relations:

(5.1) [Λ,e] = 2β, [Λ,/] = -2/, (5.2) [ej] = ^ψ^-,

eχ x χ χ where the function sh(x) is p( )-ζ p{- ) t \n this case all the finite dimensional irreducible modules are in one to one correspondence to those for the

simple Lie algebra sl2 instead of two to one for the Jimbo case with element k. We can approximate the quantum group by a family of associative algebras + J7(sl2)ib, k = 0, ••• , E Z , which are the associative algebras generated by e, /, h subject the relations (5.1) and

a* r fi - V *2i/t2m

The first of the family is the usual enveloping algebra U(sl2). Note that the commutator [e, /] is a polynomial in h of degree 2k +1. This type of algebra is studied in general in [S]. Our new algebra would have belonged to this type of algebras if there were pure commutators in the defining relations. Another feature is that our algebra is a degree 2 approximation or perturbation to the enveloping algebra. To see the behavior of our deformation one could consider another

family of algebras deviated from U(sl2) by changing the commutators in

(5.1) and (5.3) to g-commutators as in our deformation of C/(sl2). Thus one will have to generalize or g-deform the work of [S].

References

[DF] J. Ding and I. Frenkel, Quantum Clifford algebra, quantum Weyl algebra and quantum L-matrix, Yale preprint, 1992. [FRT] L. Faddeev, N. Reshetikhin and L. Takhtajan, Quantization of Lie groups and Lie algebras, Leningrad Math J., 1 (1989), 193-225. [GZ] T. Giaquinto and J. Zhang, Quantizations and deformations of Weyl algebras, Jour. Alg., 176 (1995), 861-881. m [K] G. Keller, Fusion rules of Uq(sl(2,C)), q = 1, Lett. Math. Phys., 21 (1991), 273-286. [L] L. Le Bruyn, Two remarks on Witten}s quantum enveloping algebra, U. Antwerp (UIA) preprint, 1993. [LS] L. Le Bruyn and S. P. Smith, Homogenized sh, Proc. Amer. Math. Soc, 118 (1993), 725-730. [LSV] L. Le Bruyn, S. P. Smith and M. Van den Bergh, Central extensions of'3- dimensional Artin-Schelter regular algebras, Math. Zeit., to appear. 454 NAIHUAN JING & JAMES ZHANG

[MR] J.C. McConnell and J.C. Robson, Noncommutatiυe Noetherian rings, Wiley-Inter- science, Chichester, 1987. [S] S. P. Smith, A class of algebra similar to the enveloping algebra of sl(2), Trans. Amer. Math. Soc, to appear. [WZ] J. Wess and B. Zumino, Covariant differential calculus on the quantum hyperplanes, Nucl. Phys. B (Proc. Suppl.), 18 (1990), 302-312. [W] E. Witten, Gauge theory, vertex models, and quantum groups, Nucl. Phys., B330 (1990), 285-346.

Received March 3, 1993 and revised October 6, 1993. The first author was supported by an NSA grant and the second author was supported by an NSF grant.

NORTH CAROLINA STATE UNIVERSITY RALEIGH, NC 27695-8205 E-mail address: [email protected]

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Volume 171 No. 2 December 1995

On H p-solutions of the Bezout equation 297 ERIC AMAR,JOAQUIM BRUNA FLORIS and ARTUR NICOLAU Amenable correspondences and approximation properties for von Neumann algebras 309 CLAIRE ANANTHARAMAN-DELAROCHE On moduli of instanton bundles on ސ2n+1 343 VINCENZO ANCONA and GIORGIO MARIA OTTAVIANI Minimal surfaces with catenoid ends 353 JORGEN BERGLUND and WAYNE ROSSMAN Permutation model for semi-circular systems and quantum random walks 373 PHILIPPE BIANE The Neumann problem on Lipschitz domains in Hardy spaces of order less than one 389 RUSSELL M.BROWN Matching theorems for twisted orbital integrals 409 REBECCA A.HERB

Uniform algebras generated by holomorphic and pluriharmonic functions on strictly 429 1995 pseudoconvex domains ALEXANDER IZZO Quantum Weyl algebras and deformations of U(g) 437

NAIHUAN JING and JAMES ZHANG 2 No. 171, Vol. Calcul du nombre de classes des corps de nombres 455 STÉPHANE LOUBOUTIN

On geometric properties of harmonic Lip1-capacity 469 PERTTI MATTILA and P. V. PARAMONOV Reproducing kernels and composition series for spaces of vector-valued holomorphic functions 493 BENT ØRSTED and GENKAI ZHANG Iterated loop modules and a ﬁltration for vertex representation of toroidal Lie algebras 511 S.ESWARA RAO The intrinsic mountain pass 529 MARTIN SCHECHTER A Frobenius problem on the knot space 545 RON G.WANG On complete metrics of nonnegative curvature on 2-plane bundles 569 DAVID YANG Correction to: “Free Banach-Lie algebras, couniversal Banach-Lie groups, and more” 585 VLADIMIR G.PESTOV Correction to: “Asymptotic radial symmetry for solutions of 1u + eu = 0 in a punctured disc” 589 KAI SENG (KAISING)CHOU (TSO) and TOM YAU-HENG WAN